Cycles Render - Baking texture without light influence
Ok, looks like after not quitting I have managed to find the solution.
I don't know if it is a work around for my problem or not, but it works, I did managed to add my file in the engine and has no problem, it is exactly as I was hopping to be.

Solution:
You will need to make emission instead of diffuse in the node that you will bake. Here it is a little bit more to talk how exactly I managed to bake it, but I don't wish to make lots of writings here. The solution was the above mentioned. You will need to bake from 1 object to another (by copying the original obj. with SHIFT+D) and for baking solution you will choice COMBINED (leave all the selected ones there). 

I hope that I will manage to record all this in a short tutorial and link it here.. I will need to find myself a decent desktop recorder. 
Hope that my explanation was clear enough, if not then pm me and I will gladly explain it step by step.

Blender version that I have is 2.77

Simple node settings image:
http://imgur.com/LbvQ20A

Bake image:
http://imgur.com/CtilIUu